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It follows that all vertices are congruent, ... Skilling's figure with overlapping edges. ... 2: Yes: 7: 20{3}+12 ...
The vertices can be colored with 3 colors, as can the edges, and the diameter is five. [37] The dodecahedral graph is Hamiltonian, meaning a path visits all of its vertices exactly once. The name of this property is named after William Rowan Hamilton, who invented a mathematical game known as the icosian game.
The convex regular dodecahedron also has three stellations, all of which are regular star dodecahedra.They form three of the four Kepler–Poinsot polyhedra.They are the small stellated dodecahedron {5/2, 5}, the great dodecahedron {5, 5/2}, and the great stellated dodecahedron {5/2, 3}.
The number of vertices, edges, and faces of GP(m,n) can be computed from m and n, with T = m 2 + mn + n 2 = (m + n) 2 − mn, depending on one of three symmetry systems: [1] The number of non-hexagonal faces can be determined using the Euler characteristic, as demonstrated here.
Face, a 2-dimensional element; Cell, a 3-dimensional element; Hypercell or Teron, a 4-dimensional element; Facet, an (n-1)-dimensional element; Ridge, an (n-2)-dimensional element; Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.
The vertices of Jessen's icosahedron may be chosen to have as their coordinates the twelve triplets given by the cyclic permutations of the coordinates (,,). [1] With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have length , and the long (reflex) edges have length .
v3.3.3.3.5 It is topologically related to a polyhedra sequence defined by the face configuration V4.6.2n . This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
6.6.3.3.3.3 Faces 12 E, 8 V These cannot be convex because they do not meet the conditions of Steinitz's theorem , which states that convex polyhedra have vertices and edges that form 3-vertex-connected graphs . [ 4 ]